Contents 1 OverviewandStrategies 2 SimpleApproachesandtheirDrawbacks LinearProbabilityModel Fixede ectstheIncidentalParametersProblem RandomE ectstheassumptionsaretoostrong 3 ClassicalRemedies Condi ID: 481555
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OverviewandStrategies Contents 1 OverviewandStrategies 2 SimpleApproachesandtheirDrawbacks LinearProbabilityModel Fixede ects:theIncidentalParametersProblem RandomE ects:theassumptionsaretoostrong 3 ClassicalRemedies ConditionalLogit:removingtheFixedE ects Chamberlain'sandMundlak'sApproaches:relaxingtheRandomE ectsassumption 4 Extensions Dynamicframework Semi-Parametricapproach AeberhardtandDavezies(Crest-Insee) PanelDataSeminar 11April20082/29 SimpleApproachesandtheirDrawbacks LinearProbabilityModel LinearProbabilityModel:goodforaquickstart Mainadvantage:allowstouseallthesimpleandwellknownmethodsdeveloppedforlinearmodels(FE,RE,Chamberlain'sapproach,...) Sameproblemsasinthecrosssectioncase(predictedvaluesoutsidetheunitinterval,heteroskedasticity) Evenlessappealing:itimplies�xi i1�xi AeberhardtandDavezies(Crest-Insee) PanelDataSeminar 11April200810/29 SimpleApproachesandtheirDrawbacks Fixede ects:theIncidentalParametersProblem Chamberlain'sillustrationoftheincidentalparametersproblem Verysimpleframework:MLestimationofalogitmodelwithtwoindependenttimeperiods,xede ectsandoneexplanatoryvariablexits.t.8i,xi1=0andxi2=1P(yit=1jx; )=e i+xit 1+e i+xitifyi1=0andyi2=0then^ i=�1ifyi1=1andyi2=1then^ i=+1ifyi1+yi2=1then^ i=�^=2and^=2log(~n2=~n1)P�!2with~n1=#fijyi1=1;yi2=0gand~n2=#fijyi1=0;yi1=1g AeberhardtandDavezies(Crest-Insee) PanelDataSeminar 11April200812/29 SimpleApproachesandtheirDrawbacks RandomE ects:theassumptionsaretoostrong RE:simpleprocedurebutstrongassumptions Basicassumptions: P(yit=1jxit; i)=(xit+ i) yi1;yi2;:::;yiTindependentconditionalon(xi; i) Densityof(yi1;:::;yiT)conditionalon(xi; i):f(yi1;:::;yiTjxi; i;)=TYt=1f(yitjxit; i;)=TYt=1(xit+ i)yit[1�(xit+ i)]1�yit AeberhardtandDavezies(Crest-Insee) PanelDataSeminar 11April200813/29 ClassicalRemedies ConditionalLogit:removingtheFixedE ects ConditionalLogit:makethe ivanish InthespiritofthelinearFEmodel Requiresnoassumptionon i yi1;:::;yiTindependentconditionalon(xi; i) Thedistributionof(yi1;:::;yiT)conditionalonxi; iandni=TXt=1yitdoesnotdependon i AeberhardtandDavezies(Crest-Insee) PanelDataSeminar 11April200816/29 ClassicalRemedies Chamberlain'sandMundlak'sApproaches:relaxingtheRandomE ectsassumption Strictexogeneity Allthepreviousprocedureshingeonthestrictexogeneityofxitconditionalon i: xitindependentofuit0atalltimeperiodst0 Verydiculttocorrectforendogeneityinnonlinearmodels Butaneasytestcanbeimplemented: Letwitbeasubsetofxitwhichpotentiallyfailthestrictexogeneityassumption Includewit+1asanadditionalsetofcovariates Underthenullhypothesisofstrictexogeneity,thecoecientsonwit+1shouldbestatisticallyinsignicant AeberhardtandDavezies(Crest-Insee) PanelDataSeminar 11April200820/29 Extensions Contents 1 OverviewandStrategies 2 SimpleApproachesandtheirDrawbacks LinearProbabilityModel Fixede ects:theIncidentalParametersProblem RandomE ects:theassumptionsaretoostrong 3 ClassicalRemedies ConditionalLogit:removingtheFixedE ects Chamberlain'sandMundlak'sApproaches:relaxingtheRandomE ectsassumption 4 Extensions Dynamicframework Semi-Parametricapproach AeberhardtandDavezies(Crest-Insee) PanelDataSeminar 11April200821/29 Extensions Dynamicframework ConditionalLogitinadynamicframework Youneedatleast4observationsperindividual Intuition:inordertomakethe ivanish,youneedtoconsiderthetwosetsofevents:A=fyi0=d0;yi1=0;yi2=1;yi3=d3gandB=fyi0=d0;yi1=1;yi2=0;yi3=d3g Withnoothercovariates,seeChamberlain(1985),Magnac(2000) Extensionswithstrictlyexogenouscovariates,seeHonoreandKyriazidou(2000) AeberhardtandDavezies(Crest-Insee) PanelDataSeminar 11April200823/29 Extensions Dynamicframework BacktoREframework,theinitialconditionsproblem Formofthejointdensityoftheobservationsrangingfrom0toTforanindividuali:f(yi0;yi1;:::;yiTj i;xi;)=TYt=1f(yitjyit�1;xit; i;)f(yi0jxi0; i)Goal:integratingout iinordertoobtain:f(yi0;yi1;:::;yiTjxi;)=ZTYt=1f(yitjyit�1;xit; i;)f(yi0jxi; i)g( ijxi)d iInitialconditionsproblem:specifyingf(yi0jxi; i) AeberhardtandDavezies(Crest-Insee) PanelDataSeminar 11April200824/29 Extensions Dynamicframework Initialconditionsproblem:Heckman'sapproach Specifyf(yi0jxi; i)andthenspecifyadensityfor igivenxi Forinstance,assumethatyi0followsaprobitmodelwithsuccessprobability(+xi+ i) Thenintegrateout ibyspecifyingforinstance ijxiN(mi;2i) Problem:itisverydiculttospecifythedensityofyi0given(xi; i) Problem:becausethe"true"densityofyi0given(xi; i)isnotknownandissupposedtodependonyi�1,estimatorsarebiasedwhenT+1 AeberhardtandDavezies(Crest-Insee) PanelDataSeminar 11April200825/29 Extensions Dynamicframework Initialconditionsproblem:Wooldridge'sapproach Insteadofworkingonthefulldensityf(yi0;yi1;:::;yiTj i;xi;)Wooldridgepreferstoworkontheconditionaldensityf(yi1;:::;yiTjyi0; i;xi;) Advantage:remainingagnosticonthedensityofyi0given(xi; i) Thenspecifyadensityfor igiven(yi0;xi)andkeepconditioningonyi0inadditiontoxif(yi1;:::;yiTjyi0;xi;)=Z+1�1f(yi1;:::;yiTjyi0;xit; ;)h( jyi0;xi; )d Forexample,withh( jyi0;xi; )N( +0yi0+xi;2a)yit=1f +xit+yit�1+0yi0+xi+ai+eit�0g WecanusestandardREprobitsoftwarebyjustaddingyi0andxitoalltimeperiods AeberhardtandDavezies(Crest-Insee) PanelDataSeminar 11April200826/29 Extensions Semi-Parametricapproach ReminderonManski'sapproachincrosssection(1988) Modelyi=1fxi+"i�0g Untilnow,theconditionaldensityf("jxi)wasspecied Canwerelaxthisassumption? E("jX)=0isnotenoughtoidentify(Manski,1988) med("jX)willallowtoidentify=kkunderonemoretechnicalassumptionconcerningX:theremustbeonecontinuousvariableXk,s.t.thedensityofXkjX�kispositiveeverywherea.s.0=argmaxE((2Y�1)1fX0�0g)^MS2argmaxnXi=1Yi1fX00g+(1�Yi)1fX00g AeberhardtandDavezies(Crest-Insee) PanelDataSeminar 11April200827/29